RUI: Asymptotic and Numerical Techniques in Mathematical Modeling of Membrane Filtration

膜过滤器是一种多孔介质的薄片，广泛应用于水处理、生物技术行业的各种净化工艺、肾脏透析中血液中的杂质去除、啤酒澄清和面膜生产等。膜过滤器代表着全球数十亿美元的行业，许多大型跨国公司对改进和优化他们生产的膜过滤器在性能和成本方面都保持着浓厚的兴趣。值得注意的是，实验文献远远超过了理论和数值；在理论和数值文献中，提供第一原理、预测性数学模型和模拟的研究很少。该项目旨在开发新的数学模型，有可能在弥合这一差距方面产生重大影响。长期目标是从性能和成本两方面改进和优化膜式过滤器。该项目将让学生参与研究。过滤性能强烈依赖于多孔膜的关键特性，包括膜厚度、内部孔结构和形状、孔连通性以及膜深度中孔尺寸的变化。膜形态在过滤过程中动态演变，与颗粒流动的细节(包括颗粒可能的随机行为)之间的耦合关系非常复杂，这使得过滤和污染成为一个具有挑战性的预测建模问题。这个项目提出了一个连贯的，第一原理的方法来模拟粒子动力学的随机效应和内部膜结构的变化。本研究旨在建立和分析新的数学模型，利用渐近和数值技术研究具有复杂内部结构的膜过滤器的演化过程。这些模型将与观察、实验和来自工业合作伙伴的数据进行比较，以更全面地了解在多孔介质和膜过滤器的背景下膜内部结构和流动的共同演变。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Membrane filters – thin sheets of porous medium – find widespread use in applications such as water treatment, various purification processes in the biotech industry, removing impurities from the blood in kidney dialysis, beer clarification, and mask production, among many others. Membrane filters represent a multi-billion-dollar industry worldwide, and many major multinational companies maintain a keen interest in improving and optimizing the membrane filters they produce, in terms of both performance and cost. It is notable that the experimental literature far outweighs the theoretical and numerical; among the theoretical and numerical literature, there is a paucity of studies that offer first-principles, predictive mathematical models and simulations. This project aims to develop novel mathematical models with potential for significant impact in bridging this gap. The long-term goal is to improve and optimize membrane filters, in terms of both performance and cost. The project will involve students in the research.Filter performance depends strongly on key features of the porous membrane, including membrane thickness, internal pore structure and shape, pore connectivity, and variation of pore dimensions in the depth of the membrane. The complexity of the coupling between the membrane morphology, which evolves dynamically during the filtration process, and the details of the particle-laden flow, including possible stochastic behavior of the particles, make filtration and fouling a challenging predictive modeling problem. This project presents a coherent, first-principles approach to model both stochastic effects of particle dynamics and variations in internal membrane structure. The research aims to formulate and analyze novel mathematical models to investigate the evolution of membrane filters with complex internal structures by using asymptotic and numerical techniques. These models will be compared to observations, experiments, and data from industrial partners to understand more fully the co-evolution of membrane internal structure and flow in the context of porous media and membrane filters.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.